Finding the Smallest Multiplier to Make 6627 a Perfect Square

The goal is to find the smallest positive integer that, when multiplied by 6627, results in a perfect square. A perfect square is a number that can be obtained by squaring an integer (e.g., 9 = 3², 16 = 4²).

The Role of Prime Factorization

To solve this, we need to find the prime factorization of 6627. A key property of perfect squares is that in their prime factorization, every prime factor must have an even exponent. For example, the prime factorization of 36 is 2² × 3², where both exponents are even.

Step-by-Step Prime Factorization of 6627

Let's determine the prime factors of 6627:

1. Check divisibility by 3: Sum the digits of 6627: 6 + 6 + 2 + 7 = 21. Since 21 is divisible by 3, 6627 is divisible by 3.
2. Divide by 3:

   $$ \frac{6627}{3} = 2209 $$

3. Factorize 2209: Now, we need to find the prime factors of 2209. We can test prime numbers. We find that 2209 is not divisible by primes like 5, 7, 11, etc. Upon testing larger primes, we discover that 2209 is the square of 47.

   $$ 2209 = 47 \times 47 = 47^2 $$

4. Combine the factors: The prime factorization of 6627 is:

   $$ 6627 = 3^1 \times 47^2 $$

Identifying the Required Multiplier

We examine the exponents in the prime factorization of 6627 (3¹ × 47²) to identify which factors need adjustment:

• The prime factor 47 has an exponent of 2, which is already even. This part is fine for a perfect square.
• The prime factor 3 has an exponent of 1, which is odd. To make this exponent even (specifically, 2), we need to multiply by another factor of 3.
• Therefore, the smallest number required to multiply 6627 by is 3¹, which equals 3.

Verification of the Result

Let's check if multiplying 6627 by 3 results in a perfect square:

$$ \text{New Number} = 6627 \times 3 $$

$$ \text{New Number} = (3^1 \times 47^2) \times 3^1 $$

$$ \text{New Number} = 3^{1+1} \times 47^2 $$

$$ \text{New Number} = 3^2 \times 47^2 $$

This number, 19881, is a perfect square because all exponents in its prime factorization (2 and 2) are even. It can be written as (3 × 47)² = 141².

So, the smallest number needed is 3.